# The CES Production Function

In many fields of economics, a particular class of functions called Constant Elasticity of Substitution (CES) functions are privileged because of their invariant characteristic, namely that the elasticity of substitution between the parameters is constant on their domains (for a definition of elasticity, take a glance at this post).

More production

A standard production function, linking the production factors to output is of the form:

$Y = f(L, K)$

where Y is the total production, L the quantity of human capital (labour, measured in a unit like man-hours) used and K the quantity of phyisical capital used (measured in a unit like machine-hours). Generally speaking, the production function for $n$ production factors is of the form:

$Y=f(X_1,X_2,...,X_n)$

Where $X_i$ is the quantity of factor i used. The $n$ factors generalized CES production function (also called the Armington aggregator) is:

$Y = f(X_1,X_2,...,X_n) = \left(\displaystyle\sum_{i=1}^{n}\alpha_{i}^{\rho}X_{i}^{\rho}\right)^{\frac{\gamma}{\rho}}$

With $\rho \leq 1, \rho \neq 0, \gamma > 0$. First and foremost, to study the returns to scale (refer to the link if you are not familiar with a formal definition of returns to scale) of the function, we shall study its homogeneity.

$\forall \lambda \in \mathbb{R} \backslash \{0\}: f(\lambda X_1,\lambda X_2,...,\lambda X_n) = \left(\displaystyle\sum_{i=1}^{n}\lambda^{\rho}\alpha_{i}^{\rho}X_{i}^{\rho}\right)^{\frac{\gamma}{\rho}} = \lambda^{\gamma}\left(\displaystyle\sum_{i=1}^{n}\alpha_{i}^{\rho}X_{i}^{\rho}\right)^{\frac{\gamma}{\rho}}$

In other words, the CES function is homogeneous of degree $\gamma$. Hence:

• If $\gamma > 1 \iff f(\lambda X_1,\lambda X_2,...,\lambda X_n) > \lambda f(X_1,X_2,..., X_n)$, the returns to scale are increasing.
• If $\gamma < 1 \iff f(\lambda X_1,\lambda X_2,...,\lambda X_n) < \lambda f(X_1,X_2,..., X_n)$, the returns to scale are decreasing.
• If $\gamma = 1 \iff f(\lambda X_1,\lambda X_2,...,\lambda X_n) = \lambda f(X_1,X_2,..., X_n)$, the returns to scale are constant.

The other parameters are:

• Relative weight: The $\alpha_i$ parameter associated with each production factor represents its relative distributional weight, i.e. the significance in the production.
• Elasticity of substitution: The elasticity of substitution, as the function indicates, is constant. It is: $\sigma = \dfrac{1}{1 - \rho}$ (see the addendum to this post).

What is so interesting about this function is that one can derive special cases from it:

• If $\rho \rightarrow 1$, we obtain the perfect substitutes production function: $Y = \sum_{i=1}^n\alpha_i X_i$.
• If $\rho \rightarrow -\infty$, we obtain the Leontief production function, also known as the perfect complements production function: $Y = Min \left(\dfrac{X_1}{\alpha_1},\dfrac{X_2}{\alpha_2},...,\dfrac{X_n}{\alpha_n}\right)$
• If $\rho \rightarrow 0$, we obtain the Cobb-Douglas production function, also known as the imperfect complements production function: $Y = \prod_{i=1}^n X_i^{\alpha_{i}}$.

Proving such assertions is far from trivial and my next blog post will be dedicated to deriving the Cobb-Douglas production function from the CES function.